Ligand field and density functional descriptions of the d-states and bonding in transition metal complexes

Abstract

The d-orbital energy sequences for low symmetry transition metal complexes derived from Kohn–Sham density functional theory and ligand field theory are different due to each model's treatment of interelectron repulsion. The implications for providing a unified description of the underlying metal–ligand bonding are analysed and illustrated using conventional and time-dependent DFT. Previous detailed spectroscopic studies have established the d orbital sequence in planar coordination complexes containing π-donor halide ligands as d_x²−y² ≫ d_xy > d_xz, d_yz > d_z² while for a σ-only system like [Pd(NH₃)₄]²⁺, ligand field approaches like the angular overlap model (AOM) or cellular ligand field (CLF) model predict d_xz, d_yz and d_xy should be degenerate. However, the energies of the Kohn–Sham (KS) ‘d’ orbitals of [PdCl₄]²⁻ and [Pd(NH₃)₄]²⁺ place d_xy below the d_xz/d_yz pair. Direct use of the KS orbital eigenvalues in AOM or CLF analyses would imply both ligands are π acceptors. This result is independent of the choice of functional or whether the calculation is carried out in vacuo or in a polarised continuum representing solvation by water. The origin of the difference between the KS and LFT d orbital sequences derives from their treatments of d–d interelectron repulsion. KS orbitals include interelectron repulsion contributions while LFT d orbitals do not. For a low-spin d⁸ complex, DFT gives less d–d interelectron repulsion in the xy plane leading to a lowering of d_xy relative to d_xz/d_yz. This differential effect can be reversed qualitatively by progressively removing electron density from d_z² and placing it in d_x²−y². When about 0.6 electrons is rearranged, E(d_xy) > E(d_xz/d_yz) for [PdCl₄]²⁻ and the d_π orbitals for [Pd(NH₃)₄]²⁺ are virtually degenerate. The wider ramifications of these interelectron repulsion effects are discussed for other symmetries. Excited d state energies for [PdCl₄]²⁻ are computed using both time dependent density functional theory (TDDFT) and determinant energies. The latter give the experimental sequence ¹A_2g < ¹E_g < ¹B_1g while TDDFT gives ¹E_g < ¹A_2g < ¹B_1g. Both give transition energies up to 30% lower than observed. A DFT analysis of the bonding energies in [PdCl₄]²⁻ indicates that the Pd–Cl π bonding in the molecular plane is about 33% weaker than the out-of-plane π interaction due to non-zero overlap between ligand orbitals. The normal LFT assumption of linear ligation may not always be valid.